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- Computing the metric tensor and the Riemannian manifold topology for a simple accelerating observer
Hello all,
let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##.
In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable.
We know that the Lorentz boost at every point of the world-line can be approximated with that of an O' moving with constant speed ##v(t)=at## (as ##v## increases, the ##x'## and ##t'## axes draw closer scissors-style, as it were).
If we consider this scenario from a GR point of view, the Minkowski chart is the Euclidean projection of a 2D Riemannian manifold (we might consider 1 space dimension only for the sake of simplicity), with ##x^0=t## and ##x^1=x##,
My questions are:
let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##.
In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable.
We know that the Lorentz boost at every point of the world-line can be approximated with that of an O' moving with constant speed ##v(t)=at## (as ##v## increases, the ##x'## and ##t'## axes draw closer scissors-style, as it were).
If we consider this scenario from a GR point of view, the Minkowski chart is the Euclidean projection of a 2D Riemannian manifold (we might consider 1 space dimension only for the sake of simplicity), with ##x^0=t## and ##x^1=x##,
My questions are:
- What are the values of the ##g^{\mu\nu}## as a function of ##(x,t)##, i.e. ##g^{\mu\nu}(x^0, x^1)##?
- What's the shape of the manifold embedded in the 3D space?.
- What are the functions mapping the manifold to the Euclidean 2D plane?